Shear Force and Bending Moment Diagrams

Why it matters

Shear force and bending moment diagrams are essential tools in structural analysis, helping engineers determine how beams and other structural elements will react under various loads. These diagrams are crucial for ensuring that structures can withstand applied forces without failure, which is vital for safety and reliability in construction.

Key ideas

• Shear Force (SF): The internal force parallel to the cross-section of a structural element. It results from external loads, reactions at supports, or changes in geometry.
• Bending Moment (BM): The internal moment that causes a beam to bend. It is the result of external loads applied at a distance from a section.
• Sign Conventions: Here positive bending is sagging; use the associated section convention with dM/dx = V and dV/dx = -w for downward w.
• Diagrams: Graphical representations showing how shear force and bending moment vary along the length of a beam.
• Types of Loads: Point loads, distributed loads, and moments can all affect shear force and bending moment.

Formulas

• V = ΣFy
  • V: Shear force (N)
  • ΣFy: Sum of vertical forces (N)
• M = Σ(Mo)
  • M: Bending moment (Nm)
  • Σ(Mo): Sum of moments about a point (Nm)

Worked example

Given: A simply supported beam of length 6 m with a point load of 10 kN at the center.

1. Calculate reactions at supports.

   • Assume reactions at supports A and B are RA and RB.
   • RA + RB = 10 kN
   • Taking moments about A: RB * 6 m = 10 kN * 3 m
   • RB = 5 kN
   • RA = 10 kN - 5 kN = 5 kN

2. Draw shear force diagram.

   • From A to C (center): V = 5 kN
   • At C, V = 5 kN - 10 kN = -5 kN
   • From C to B: V = -5 kN

3. Draw bending moment diagram.

   • At A and B, M = 0
   • At C, M = 5 kN * 3 m = 15 kNm

Final Answer: Maximum bending moment is 15 kNm at the center.

Common mistakes

• Incorrect sign conventions leading to wrong diagrams.
• Miscalculating reactions at supports.
• Forgetting to consider all types of loads, including distributed loads.

For GATE CE

Questions often involve calculating shear force and bending moment at specific points or drawing the entire diagram for a given beam. Practice with different loading conditions and beam configurations to master this topic.

Quick check

1. What is the shear force at the midpoint of a simply supported beam with a central point load?
2. How does a uniformly distributed load affect the bending moment diagram?
3. What is the bending moment at the supports of a simply supported beam?

Answers: 1. There is a jump from +P/2 on the left to -P/2 on the right; there is no unique zero shear at the point load, 2. It creates a parabolic curve, 3. Zero.